Problem 30 - Elementary and Intermediate Algebra

Question

Use the product rule for exponents to simplify each expression. Write the results using exponents. $$ a a^{3} a^{5} $$

Step-by-Step Solution

Answer

The simplified expression is \( a^9 \).

Step 1: Identify the Expression

The given expression is \( a \cdot a^3 \cdot a^5 \). We need to use the product rule for exponents to simplify this.

Step 2: Recall the Product Rule for Exponents

The product rule for exponents states that when multiplying like bases, you add the exponents: \( a^m \cdot a^n = a^{m+n} \).

Step 3: Apply the Product Rule

Use the product rule to combine the exponents for each \( a \) in the expression: \( a^1 \cdot a^3 \cdot a^5 = a^{1+3+5} \).

Step 4: Simplify the Expression

Add the exponents: \( 1 + 3 + 5 = 9 \). So the simplified expression is \( a^9 \).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra. It involves reducing algebraic expressions to their simplest form. This makes calculations easier and helps identify the core relationships in mathematical problems. Consider the expression \( a \cdot a^3 \cdot a^5 \). Each term includes the base \( a \). The goal of simplifying this expression is to express it as a single term with one exponent.

• First, identify terms that share the same base. Here, each term is a power of \( a \).
• Next, apply rules such as the product rule for exponents, which makes complexity manageable.
• Address each like term, combining them efficiently to simplify the expression.

Breaking down expressions allows us to view complex problems as solvable, step-by-step tasks.

Exponents in Algebra

Exponents in algebra are used to denote repeated multiplication of the same factor. They are crucial in defining how many times a number, known as the base, is multiplied by itself. In the given problem, \( a^3 \) and \( a^5 \) represent powers of the base \( a \). Understanding their role is key to mathematical fluency.

• An exponent of 1, \( a^1 \), means the base is used once, equivalent to just \( a \).
• In the product \( a^3 \cdot a^5 \), each part is a distinct power of \( a \), showing separate sequences of multiplication.
• Exponents simplify expressing large numbers or complex problems by using a compact form.

Recognizing patterns within exponents forms the foundation for manipulating algebraic expressions effectively.

Mathematical Properties

Mathematical properties, such as the product rule, allow us to manipulate expressions with efficiency and precision. The product rule specifically aids in multiplying exponents with the same base.

• The product rule states: \( a^m \cdot a^n = a^{m+n} \), where \( a \) is the base, and \( m \) and \( n \) are exponents.
• This rule simplifies processes by converting multiplication into addition of exponents.
• It is particularly useful in algebra when dealing with polynomials or equations involving powers.

Exploiting these rules makes solving complex equations quicker and prevents errors. Embracing mathematical properties like the product rule is essential for efficient problem-solving in mathematics.